where \(p(x)\)  captures some newly obtained information. (The notation convention follows [1]}: \(q(x)\)  denotes priors, while \(p(x)\)  denotes a possible candidate posterior, and \(P(x)\)  signifies the optimal posterior.) The strategy we implement for accomplishing this is the following: we rank the possible posteriors \(p(x)\)  according to preference. (We will be more specific about what we mean by preference shortly.) That is, to each possible posterior distribution \(p_{i}(x)\)  we assign a single real number \(S[p_{i},q]\equiv S[p_{i}]\)  — the entropy functional — such that if we prefer \(p_{1}\)  to \(p_{2}\)  then \(S[p_{1}] > S[p_{2}]\) .The ranking scheme that we utilize for updating probability distributions can traces its origins to the work of Shore and Johnson [2]} as well as J. Skilling [3]}. Naturally, such a ranking, if it is to be useful, should also be transitive: if \(p_{1}\)  is preferred to \(p_{2}\)  — \(S[p_{1}] > S[p_{2}]\)  — and \(p_{2}\)  preferred to \(p_{3}\)  — \(S[p_{2}] > S[p_{3}]\)  — then \(p_{1}\)  is preferred to \(p_{3}\)  — \(S[p_{1}] > S[p_{3}]\) .
